[[Magmoid]]
# Magmoid homomorphism

A **magmoid homomorphism** $F : \mathscr M \to \mathscr N$ is a structure-preserving map between [[Magmoid|magmoids]]. 
It associates:

- An object $FX \in \mathscr N$ for every $X \in \mathscr M$
- An arrow $F\alpha \in \mathscr N(FX, FY)$ for every $\alpha \in \mathscr  M(X,Y)$

such that $(F\alpha) \odot (F\beta) = F(\alpha \odot \beta)$ for any $\alpha \in \mathscr M(X,Y)$ and $\beta \in \mathscr M(Y,Z)$.

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