# Composition of functors

I'm having trouble presenting a proof for the intuitive fact that the composition of two full functors is again full.

Mostly, I'm having trouble doing the symbolic transformations that get me to the desired result.

I would like some advice on how to tackle this question.

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welcome to math SE Allison! –  magma Oct 26 '12 at 1:31

A functor is full if every map $g : FX \to FY$ is equal to $Ff$ for some $f : X \to Y$.
The functor $FG$ is full when $F$ and $G$ full since, using fullness of $F$, we can express $g : FGX \to FGY$ as $Fh$ for some $h : GX \to GY$ which, using fullness of $G$, we can express as $Gf$ for some $f : X \to Y$. Putting both those facts together we can express $g : FGX \to FGY$ as $FGf$.