The category of monoids on $\smash {\webleft (\mathsf{Sets}_{*},\rhd ,S^{0}\webright )}$ is isomorphic to the category of “monoids with right zero”1 and morphisms between them.
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The category of monoids on $\smash {\webleft (\mathsf{Sets}_{*},\rhd ,S^{0}\webright )}$ is isomorphic to the category of “monoids with right zero”1 and morphisms between them.
determining a right bilinear morphism of pointed sets
picking an element $1_{A}$ of $A$.
commutes.
commutes.
Being a right-bilinear morphism of pointed sets, the multiplication map satisfies
for each $a\in A$. Now, the associativity, left unitality, and right unitality conditions act on elements as follows:
for each $a,b,c\in A$.
for each $a\in A$.
This gives
for each $a\in A$.
Thus we see that monoids with respect to $\rhd $ are exactly monoids with right zero.
satisfying the following conditions:
commutes.
commutes.
These act on elements as
and and giving
for each $a,b\in A$, which is exactly a morphism of monoids with right zero.