Creating tikz picture | Problem with arrows and labels

Question

I am pretty new on TikZ and I want to create a diagram for my measure class. I created the following picture:

\documentclass{standalone}
\usepackage{float}
\usepackage{amsfonts}
\usepackage{mathrsfs}
\usepackage{tikz}
\begin{document}
   \begin{tikzpicture}[scale=1.5]
        %Mis espacios
        \node (A) at (0,3) {$(\Omega, \mathscr{F},\mu)$};
        \node (B) at (3,3) {$(\Omega',\mathscr{F}')$};
        \node (C) at (3,0) {$(\mathbb{R}^1,\mathscr{R}^{1})$};
        %Los mapas
        \draw [thick,->] (A) to  node[midway,above] {$T$} (B);
        \draw [thick,->] (B) to node[midway,right] {$f$} (C);
        \draw [dashed,thick,->] (A) to node[midway,above left, yshift=-13 pt] {$T \circ f$} (C);
    \end{tikzpicture}
\end{document}

However I am having problems with the executions of the code since the label of compositions appears over the arrow and the arrow tips do not appear.

The following code includes all the packages I have originally in case some package is not compatible with my picture.

%Colores
\usepackage{xcolor}
%Evita problemas con tildes y caracteres especiales
\usepackage[utf8]{inputenc}
%Secciones con formato
\usepackage{sectsty}
%Formato del Documento
\usepackage[left=2.5cm,top=2.5cm,right=2.5cm,bottom=2.5cm]{geometry}
%Formato de entornos de enunciados matemáticos
\usepackage{amsthm}
%amsmath + más simbolos utiles
\usepackage{mathtools}
%Cosas en Español
\usepackage[spanish]{babel}
%Hipervínculos
\usepackage{hyperref}
%Letras de conjuntos notables
\usepackage{dsfont}
%Más simbolos matemáticos útiles
\usepackage{amssymb}
%Inclusión de imágenes
\usepackage{graphicx}
%Hipervínculos
\usepackage{hyperref}
%Comentarios largos
\usepackage{comment}
%Citas y Referencias
\usepackage{apacite}
%Editando las etiquetas de las numeraciones
\usepackage{enumitem}
%Caritas
\usepackage{wasysym}
%Interlineado
\usepackage{setspace}
%tablas acomodadas 
\usepackage{float}
%Rotar en Tabla 
\usepackage{multirow}
%Encabezado y pie de página
\usepackage{fancyhdr}
%cancelaciones
\usepackage{cancel}
%Letras caligraficamente perfectas
\usepackage{mathrsfs}
%Graficas poderosa
\usepackage{tikz}

Also, I would like to include the error that appears on my original document:

Argument of \language@active@arg> has an extra }.
Runaway argument?
Missing \endcsname inserted.

Answer 1

Please find attached a few choices you have. You may want to look them up in the pgfmanual, e.g. via index, find etc. I indicated changes in the code, both by \section and some visual comment. Option transform shape together with scale was probably the one you were missing.

% https://tex.stackexchange.com/questions/684274/creating-tikz-picture-problem-with-arrows-and-labels?noredirect=1#comment1697768_684274

%\documentclass{standalone}
\documentclass[10pt]{article}
%\usepackage{float}% <<<--- not needed here
\usepackage{amsfonts}
\usepackage{mathrsfs}
\usepackage{tikz}

\begin{document}

\section{as posted}
   \begin{tikzpicture}[scale=1.5]
        %Mis espacios
        \node (A) at (0,3) {$(\Omega, \mathscr{F},\mu)$};
        \node (B) at (3,3) {$(\Omega',\mathscr{F}')$};
        \node (C) at (3,0) {$(\mathbb{R}^1,\mathscr{R}^{1})$};
        %Los mapas
        \draw [thick,->] (A) to  node[midway,above] {$T$} (B);
        \draw [thick,->] (B) to node[midway,right] {$f$} (C);
        \draw [dashed,thick,->] (A) to node[midway,above left] {$T \circ f$} (C);
    \end{tikzpicture}

\section{changed scale}    
    \begin{tikzpicture}[scale=1.0]% <<<---
        %Mis espacios
        \node (A) at (0,3) {$(\Omega, \mathscr{F},\mu)$};
        \node (B) at (3,3) {$(\Omega',\mathscr{F}')$};
        \node (C) at (3,0) {$(\mathbb{R}^1,\mathscr{R}^{1})$};
        %Los mapas
        \draw [thick,->] (A) to  node[midway,above] {$T$} (B);
        \draw [thick,->] (B) to node[midway,right] {$f$} (C);
        \draw [dashed,thick,->] (A) to node[midway,above left] {$T \circ f$} (C);
    \end{tikzpicture}


\section{with shorten}
   \begin{tikzpicture}[scale=1.5]
        %Mis espacios
        \node (A) at (0,3) {$(\Omega, \mathscr{F},\mu)$};
        \node (B) at (3,3) {$(\Omega',\mathscr{F}')$};
        \node (C) at (3,0) {$(\mathbb{R}^1,\mathscr{R}^{1})$};
        %Los mapas
        % <<<--- -----------|
        \draw [thick,->,shorten >=10pt] (A) to  node[midway,above] {$T$} (B);
        \draw [thick,->,shorten >=10pt] (B) to node[midway,right] {$f$} (C);
        \draw [dashed,thick,->,shorten >=10pt] (A) to node[midway,above left] {$T \circ f$} (C);
    \end{tikzpicture}
    

\section{with transform shape}
   \begin{tikzpicture}[scale=1.5, transform shape]% <<<---
        %Mis espacios
        \node (A) at (0,3) {$(\Omega, \mathscr{F},\mu)$};
        \node (B) at (3,3) {$(\Omega',\mathscr{F}')$};
        \node (C) at (3,0) {$(\mathbb{R}^1,\mathscr{R}^{1})$};
        %Los mapas
        \draw [thick,->] (A) to  node[midway,above] {$T$} (B);
        \draw [thick,->] (B) to node[midway,right] {$f$} (C);
        \draw [dashed,thick,->] (A) to node[midway,above left] {$T \circ f$} (C);
    \end{tikzpicture}


\section{with transform shape and shifted text}
   \begin{tikzpicture}[scale=1.5, transform shape]% <<<---
        %Mis espacios
        \node (A) at (0,3) {$(\Omega, \mathscr{F},\mu)$};
        \node (B) at (3,3) {$(\Omega',\mathscr{F}')$};
        \node (C) at (3,0) {$(\mathbb{R}^1,\mathscr{R}^{1})$};
        %Los mapas
        \draw [thick,->] (A) to  node[midway,above] {$T$} (B);
        \draw [thick,->] (B) to node[midway,right] {$f$} (C);
        % <<<--- ------------------------------------------------|
        \draw [dashed,thick,->] (A) to node[midway,above left,yshift=-1.5em] {$T \circ f$} (C);
    \end{tikzpicture}
   
\end{document}

Answer 2

I found that the language pack interferes with the use of the > character in the arrows so you can try using another type of definition as offered by normal tikz or you can use the meta arrows that is in the example case. I recommend you see the tikz visual manual pages 21 and 56.

RESULT:

MWE:

\documentclass[tikz]{standalone}
\usepackage[utf8]{inputenc}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage[spanish]{babel} %<---Here this throws an error using the ">" symbol for arrows, try using arrows.meta
\usepackage{mathrsfs}
\usepackage{float}
\usepackage{amsfonts}
\usepackage{mathrsfs}
\usetikzlibrary{arrows.meta} %<--- 
\begin{document}

   \begin{tikzpicture}[]
        %Mis espacios
        \node (A) at (0,3) {$(\Omega, \mathscr{F},\mu)$};
        \node (B) at (3,3) {$(\Omega',\mathscr{F}')$};
        \node (C) at (3,0) {$(\mathbb{R}^1,\mathscr{R}^{1})$};
        %Los mapas
         \draw [thick,-Classical TikZ Rightarrow] (A) to  node[midway,above] {$T$} (B);
        \draw [thick,-To] (B) to node[midway,right] {$f$} (C);
        \draw [dashed,thick,-Triangle] (A) to node[midway,sloped,above] {$T \circ f$} (C);
    \end{tikzpicture}
\end{document}

By the way, I did not find any other incompatibility with the other packages that you list, which of course are required for your main document, in the standalone tikz environment it could be declared at the beginning and tikz itself invokes others like xcolor, certainly if it is a The final image should be managed in your main document as if it were a normal image, only that you must import it in pdf format, so you can scale it, rotate it, even nest it, you can see that in the following post from a long time ago: Scaling different components of tikzpicture together

Answer 3

Using tikz-cd package.

\documentclass[a4paper,12pt]{article}
\usepackage{amsmath,amssymb}
\usepackage{tikz-cd}
\usepackage{mathrsfs}

\begin{document}
\begin{tikzcd}[row sep =1.5cm]
{(\Omega, \mathscr{F},\mu)} \arrow[rd, "T\circ f"{sloped}, dashed] \arrow[r, "T"] & {(\Omega',\mathscr{F}')} \arrow[d, "f"] \\
& {(\mathbb{R}^1,\mathscr{R}^{1})}       
\end{tikzcd}
\end{document}