Container Homomorphism

A container homomorphism is a structure preserving map between Container. Given containers $S◁P;T◁Q$, a homomorphisms must be of the form,

$$\sum _{f:S\to T}\prod _{x:S}Q(fx)\to Px$$

It can be interpreted as a map between shapes $f$, and a backward map $\sigma$ from the resulting positions $Q(fx)$ to positions in the prior position for each shape $x$.