Given a pair of functors $F,G : \mathbf{Set} \rightarrow \mathbf{Set}$, we get a category as follows:
Objects. Sets $S$ together with a function $\hat{S}:FS \rightarrow GS$
Arrows. A morphism $f : S \rightarrow T$ is just a function between the relevant underlying sets, such that $F(f) \circ \hat{S} = \hat{T} \circ G(f)$.
For example:
if $X$ and $Y$ are sets and we choose $F = X \times -$ and $G = Y$, then we obtain a definition of $Y^X$ as the terminal object in the relevant category.
if $F = 1+-$ and $G=-$ (i.e. $G$ is the identity), then we obtain a definition of $\mathbb{N}$ as the initial object in the relevant category.
Question. Is it a general principle that all categories obtained in this way have initial and terminal objects?
Addendum. As Nex explains, the answer is no: take $F = 1$ and $G=0$, then the above category as no objects. So, I'd be interested in conditions under which initial and terminal objects exist.