Remembering arrows' directions in basic Category Theory

I like to think of such properties in the forms they take for sets. For instance, I remember being rather surprised that the fibered product was actually a rather familiar thing (ordered pairs that lie over the same point in the base) after I read the section on it in Hartshorne, which just talked about the universal property, Yoneda's lemma means that you can restrict yourself to the case of Sets in proving results about them (Grothendieck spends some time on this in EGA 0 and gives plenty of detail).

For co things, though, you have to hom out of them. A map out of a quotient A/B is the same thing as a map out of A that vanishes on B. So, more generally, maps out of co things satisfy nice properties.