graphics - Creating diagrams for category theory

Lately I've been doing algebra and I have found myself drawing a bunch of diagrams when I attempt to solve a problem. Most of the diagrams are very simple so I thought, I bet I can do this in Mathematica and include these pictures in a nice document. What I would like to do is create diagrams like the ones found in category theory:

Here is an attempt based on a solution provided by Mr.Wizard

vertex = {
    {White, Disk[#2, 0.04]},
    Text[#, #2]

    } &;
GraphPlot[{
   {"A" -> "B", "f"},
   {"A" -> "C", "h"},
   {"B" -> "C", "g"}
   },
  DirectedEdges -> True,
  VertexLabeling -> True
  ] /. {Text -> vertex, Framed -> (# &)}

I like the idea of how in GraphPlot you can define the vertices and the labels. Unfortunately I'm still not very savvy on controlling many aspects of the Graphics. Does anyone know how to create diagrams like the ones I'm attempting to draw? It would be awesome if the font size could be specified.

Edit:

As belisarius pointed out:

Text length and font size should be considered with care

As requested by Mr.Wizard, here is are some interesting diagrams:

https://scientopia.org/blogs/goodmath/category/good-math/category-theory/

The second picture can be found here

A quick search on google images for "category theory diagrams" will give you some other diagrams including the ones I have put in here. It seems based on the figures that most of them involve simple rectangles with diagonal lines. In some cases some lines might be curved but this can all be done with straight lines.

Since the arrangement of the lines is up to personal choice it would be good to have a solution in which we can move the vertices locations, a similar functionality should be done with the edge labels. I picture an edge label as an image that can rotate around a point in a line. That way we can select a position in the line and the position angle. In most cases this angle is multiples of 45 degrees.