Subobject classifier in $Set^{C^{op}}$ I'm reading "Sheaves in geometry and logic" and I'm not sure if i'm understanding some definitions.

We have our functor $\Omega$ defined on objects by $\Omega(C)$$=\{$$S|$ $S$ is a sieve on C in $\mathcal{C}$ $\}$ and on arrows $g: C^{'} \rightarrow C$ by
$\Omega(f)$: $S \rightarrow S·g$
such that $S·g=$$\{$$h|$ $g\circ h$$\in S$$\}$
We define $true: 1\rightarrow \Omega$ sending each $C \in \mathcal{C}$ to the maximal sieve on $C \in \Omega(C)$
Then if $Q$ is a subfunctor of $P$ we've got the map $\phi_c (x)$$=$$\{$$f|$$ $$x·f\in Q(dom(f)) \}$ where $f$ ranges over all morphism in $\mathcal{C}$ with codomain C

I don't  get what $x·f$ does mean since $P$ doesn't have to be a representable functor so x is not really a function which can be composed.

Generalizing the examples of groups acting on a set or rings acting on abelian groups, we can consider categories acting on sets. The simplest description is that any functor $F : \mathcal{C} \to \mathrm{Set}$ gives a left action of $\mathcal{C}$, and $G : \mathcal{C}^\circ \to \mathrm{Set}$ gives a right action.
And following the examples, in the case of a right action we would notation $x \cdot f$ to denote the result of applying the function $G(f)$ to the element $x$.
