# Push-outs of fully faithful (enriched) functors

I'd like to know a reference where the following property, that I believe to be true, is checked: given a diagram of categories and functors $B\leftarrow A\rightarrow C$ (I'm actually interested in the enriched case, over a closed symmetric monoidal category), if $A\rightarrow C$ is fully faithful then so is the functor to the (strict) push-out $B\rightarrow B\cup_A C$.

I should say that I have been able to check this in some specific cases that I needed, but a reference would help me to reduce a paper in four pages ;-)

• @Adeel, thanks, I knew that paper, it says that for ordinary categories $B\r B\cup_AC$ is an inclusion. That's a hint, in my opinion, about the fact that my guess is true. I'd like it to be also full, and for enriched categories. – Fernando Muro Jan 22 '14 at 17:39
• On the other hand, the proofs in the cited paper heavily use the explicit combinatorial description of morphisms in the pushout, so it seems unlikely to me that this will generalise to the enriched case. – Zhen Lin Jan 23 '14 at 9:53
• @ZhenLin, I think it does, everything can be done diagramatically, via tensor products and push-outs. I hope somebody has done something like that before! Or maybe there is a short argument that I cannot find. – Fernando Muro Jan 23 '14 at 10:12
• You may be interested in contributing to a proposal Spanish language version of math stackexchange; it could use some input from fluent professors: area51.stackexchange.com/proposals/64529/… – Brian Rushton Feb 1 '14 at 3:13