If $g \circ f$ is monic, then $f$ is monic. Or, if $g \circ f \circ h = g \circ f \circ k \implies h = k$, then $ f \circ h = f \circ k \implies h=k$.
I am not exactly sure how to prove this. I don't know what my possible actions or manipulations are to get from one statement to the other. I drew a (messy) commuting diagram (no idea how to do it in LaTeX), where everything commutes, but I don't know how to turn that into a proof.
The commuting diagram looked like this: a square with h on top, k on the left, and $g\circ f$ on the bottom and right - this square commutes by the assumption - then I added in the composition triangles for each $g\circ f$, identifying the codomain of f in the middle of the square, making it look like a pushout diagram. This makes a new square with h on top, k on the left, and f in the bottom and right, the commuting diagram for a monomorphism. Everything else in the diagram commutes. So I guess a second question, does that mean the square commutes? (A diagram commutes if every triangle in it commutes, is the reverse true?)