What does it mean to be functorial (in something)? Is there a definition of the word functorial? Often I read that certain things are functorial and often I have an intuition of what is meant. But is there a precise definition of what is meant by this phrase or do you have to guess each time? Of course the more experienced people in category theory will immediately know when they read a sentence using the word functorial which releation is meant but for me it's sometimes difficult and I'm not sure if I guess correct or if the author actually meant a deeper relation.
Here two examples where the word functorial is used in case they help you to give a definition:
Example 1: Let $\mathcal F$ be a sheaf on $X$ and $f:X\to Y$, $g:Y\to Z$ be continuous maps. Then $g_*(f_* \mathcal F) = (g \circ f)_* \mathcal F$ is functorial in $\mathcal F$.
Example 2: Let $\mathcal C$ be a category whose hom-sets themselves carry the structure of a category such that composition is functorial.
 A: It's actually pretty straightforward and your intuitions are probably right.
Basically, when you have some expression $E$ that contains $F$, and we say $E$ is functorial in $F$, it means that $F \mapsto E$, is a functor.  In your first example it's saying that $\mathcal{F}\mapsto g_* (f_* \mathcal{F})$, i.e. $g_* f_*$, and $(g \circ f)_*$ are both functors and that they are equal as functors.  This is very similar to how we might say some transformation is natural in some objects, e.g. we say "$F \dashv U$ means $\text{Hom}(FX,Y)\cong\text{Hom}(X,UY)$ natural in $X$ and $Y$."  In fact, this statement implies that each side is functorial in $X$ and $Y$.  There is a bit of ambiguity here (and in general) as it can be ambiguous which domain category is intended given only the action on objects.  For example, it takes knowing $X \in \text{Ob}(\mathcal{C}^{op})$ to see that this means $(X,Y) \mapsto \text{Hom}(FX,Y) : \mathcal{C}^{op}\times\mathcal{D} \to \mathbf{Set}$.  Usually the intended category is clear enough.
The composition example can be cast the same way. Using $\mathbf{Cat}$ as the archetypal 2-category for concreteness, we have between any two (1-)categories $\mathcal{C}$ and $\mathcal{D}$, the functor (1-)category $[\mathcal{C},\mathcal{D}]$. For $F:[\mathcal{D},\mathcal{E}]$ and $G:[\mathcal{C},\mathcal{D}]$, the expression $F \circ G$ is functorial in $F$ and $G$.  That is, $(F,G) \mapsto F \circ G : [\mathcal{D},\mathcal{E}]\times[\mathcal{C},\mathcal{D}] \to [\mathcal{C},\mathcal{E}]$ is a functor.  It's action on arrows is just horizontal composition of natural transformations in this case.
A: Usually the word functorial means that some construction of some sort is a functor. 
There are two example of this notion that come to my mind:


*

*if you have a graph-map between the graphs underlying two categories, such map is functorial if it verifies the condition for being a functor: namely it preserves composites and identities morphisms

*if you have a function between the collections of objects of two categories, then this mappings is said to be functorial the object-function can extened to a functor by defining a mapping between the collections of morphisms, which is compatible with composition, identities and source and target operations.


I am not sure how the first example fall in one of these two cases, because I am not really familiar with the notations you use.
Surely the second example falls in the second case: if $\mathcal C$ is a category such that for each pair of objects $A,B \in \mathcal C$ the set $\mathcal C[A,B]$ is the set of the objects for a category $\bar {\mathcal C}[A,B]$, then it is natural to require that the compositions operations 
$$\circ \colon \mathcal C[B,C] \times \mathcal C[A,B] \longrightarrow \mathcal C[A,C]$$
can extended to a family of functors 
$$\overline \circ \colon \overline{\mathcal C}[B,C] \times \overline{\mathcal C}[A,B] \longrightarrow \overline{\mathcal C}[A,C]\ .$$
Hope this helps.
Edit: On second thoughts I believe that also the first example you gave fall in the first case: I assume that example 1 means to express that the mapping $\mathcal F \mapsto (g \circ f)_*(\mathcal F)$ that sends a sheaf of $X$ into a sheaf of $Y$ (that is a function between the collection of the objects of the sheaf-categories $\mathbf{Sh}(X)$ e $\mathbf {Sh}(Y)$) can be extended to a functor (by defining the actions of $(g \circ f)_*$ on the morphisms of sheaves in the obvious way).
