Let $F:\mathcal{A}\to\mathcal{A}$ be a functor with a choice of $A\to F(A)$ for every $A\in \operatorname{Ob}\mathcal{A}$, such that
$$\require{AMScd} \begin{CD} A @>{f}>> B\\ @VVV @VVV \\ F(A) @>{F(f)}>> F(B). \end{CD}$$
In other words, there is a natural transformation $\eta:Id\to F$. I'm wondering if these kind of functors have important properties that I should know of.
On
The wonderful "and so on" paper by Kelly answers almost any possible question on the subject of pointed functors:
Max Kelly, “A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on.” Bull. Austral. Math. Soc. 22 (1980), 1–83.
A dry account of what Kelly does: in the lab, where else?
Well, for one, these functors are closed under composition; for example, if you have natural transformations $\tau : \mathbf 1 \to F, \eta : \mathbf 1 \to G$, you get natural transformations $\mathbf 1 \to FG$ and $\mathbf 1 \to GF$. You have natural transformations $\mathbf 1 \to F^n$ for all $n$.
I like to think of a natural transformation $\mathbf 1 \to \mathcal F$ as providing a small deformation of the category $\mathcal A$, with the following analogy. If $X$ is a manifold, and $f$ an endomorphism of $X$, a homotopy $\mathbf 1_X \to f$ can be thought of as coherent family of paths from $x$ to $f(x)$, indexed by $x \in X$. The path starting at $x$ gives a tangent vector at $x$, and so we get a vector field on $X$.
You can think of the data $(F, \tau)$ as a procedure which transforms each object $A$ into an object $F(A)$, into which $A$ maps canonically.
This is also part of the data defining a monad.