Motivation. Piggybacking a previous question that I posted, and drawing the colimit diagram for reference and notation:
I wanted to be able to formally say that $\phi_{X},\phi_{Y}$ were injections (in the usual sense) so that, for example, taking a pushout would give me the "natural" pushout (as opposed to some other colimit using different $\phi_{X}$ and $\phi_{Y}$ mappings which are not the injection mappings). In some cases, I would explicitly be in some category (like spaces) and I could say something like $\phi_{X}(y) = y$ in order to force my map to be the injection map; but in other cases where the arrows between objects are not functions (like poset categories) I'm not sure how to say this formally.
Question. When one talks about taking "the usual" pushout, how does one define the injective arrows $\phi_{X}$ and $\phi_{Y}$ for categories where the arrows are not functions (as in the category of posets).
I'm a bit new at this stuff, so if my question is not well-stated or garbled I will attempt to reword it!