Complicated diagram with xy (everything is wrong)

Question

I'm trying to create a diagram for the countable direct limit in algebra. The following code

$$\vspace{-1mm}\xymatrix{
&&& A\ar@{-->}[dd]^{\exists!\alpha}\\
A_1\ar[r]^{\!\!\!\alpha_{1,2}} \ar@/^12pt/[urrr]|-{\alpha_{1}} \ar@/_12pt/[drrr]|-{\alpha'_{1}}&
A_2\ar[r]^{\!\!\!\alpha_{2,3}} \ar[urr ]|-{\alpha_{2}} \ar[drr ]|-{\alpha'_{2}}&
A_3\ar[r]^{\!\!\!\alpha_{3,4}} \ar[ur  ]|-{\alpha_{3}} \ar[dr  ]|-{\alpha'_{3}}&
\ldots\\
&&& A'\\}$$

produces the diagram below:

Questions:

1. How can I make the arrows \alpha_1,\alpha_2,\alpha_3 arrive at their destination A?

2. How to put labels \alpha_{1,2} lower (closer to their arrow)?

3. How to move objects A and A' a bit to the right (along with \alpha_i,\alpha'_i) so that the arrow \alpha doesn't intersect \ldots and remains straight?

4. When I bend arrows (using PDFtex) I a get rasterized arrow. How can I bend arrows and still have vector graphics?

5. What is wrong with \alpha'_3?

Answer 1

I would bend also the dashed arrow and also, slightly, the second diagonal arrows:

\[
\xymatrix{
&&& A\ar@/^12pt/@{-->}[dd]^{\exists!\alpha}\\
A_1\ar[r]^-{\alpha_{1,2}}
\ar@/^12pt/[urrr]|-{\alpha_{1}}
\ar@/_12pt/[drrr]|-{\alpha'_{1}}&
A_2\ar[r]^-{\alpha_{2,3}}
\ar@/^4pt/[urr]|-{\alpha_{2}} \ar@/_4pt/[drr]|-{\alpha'_{2}}&
A_3\ar[r]^-{\alpha_{3,4}} \ar[ur]|-{\alpha_{3}} \ar[dr]|-{\alpha'_{3}}&
\ldots\\
&&& A'
}
\]

Note that \ar[r]^-{f} puts the label in the middle of the actual arrow.

A solution adding a column for the dots is not as good.

Answer 2

Remove the spaces in the sqare brackets for \ar. This solves problems 1 and 5.

\documentclass{article}
\usepackage[all]{xy}
\begin{document}
\[ 
\vspace{-1mm}\xymatrix{
&&& A\ar@{-->}[dd]^{\exists!\alpha}\\
A_1\ar[r]^{\!\!\!\alpha_{1,2}} \ar@/^12pt/[urrr]|-{\alpha_{1}} \ar@/_12pt/[drrr]|-{\alpha'_{1}}&
A_2\ar[r]^{\!\!\!\alpha_{2,3}} \ar[urr]|-{\alpha_{2}} \ar[drr]|-{\alpha'_{2}}&
A_3\ar[r]^{\!\!\!\alpha_{3,4}} \ar[ur]|-{\alpha_{3}} \ar[dr]|-{\alpha'_{3}}&
\ldots\\
&&& A'\\}
\]
\end{document}