We consider the category of endofunctors of finite sets with natural transformations.

Is there an infinite chain $F_1, F_2, \ldots$ of endofunctors such that there is a natural transformation from $F_i$ to $F_{i+1}$ for all $i\in \mathbb{N}$ but no natural transformation from $F_{i+1}$ to $F_i$ for any $i\in \mathbb{N}$?

(This problem has a relevance in universal algebra as one can consider for each pair of finite structures $A, B$ the functor mapping the set $\{ 1,2,...,n\}$ to the set of homomorphisms $\operatorname{Hom}(A^n,B)$ with the obvious maps on morphisms. In fact, this was the motivation for the question.)