Let $A$ be a set.
A reflexive relation is equivalently:1
An $\mathbb {E}_{0}$-monoid in $(\mathrm{N}_{\bullet }(\mathbf{Rel}(A,A)),\chi _{A})$.
A pointed object in $(\mathbf{Rel}(A,A),\chi _{A})$.
In detail, a relation $R$ on $A$ is reflexive if we have an inclusion
of relations in $\mathbf{Rel}(A,A)$, i.e. if, for each $a\in A$, we have $a\sim _{R}a$.
Let $A$ be a set.
The set of reflexive relations on $A$ is the subset $\smash {\mathrm{Rel}^{\mathrm{refl}}(A,A)}$ of $\mathrm{Rel}(A,A)$ spanned by the reflexive relations.
The poset of relations on $A$ is is the subposet $\smash {\mathbf{Rel}^{\mathsf{refl}}(A,A)}$ of $\mathbf{Rel}(A,A)$ spanned by the reflexive relations.
Let $R$ and $S$ be relations on $A$.
To prove that $a\sim _{S\mathbin {\diamond }R}a$ holds as well, we have to exhibit some $b\in A$ such that $a\sim _{R}b$ and $b\sim _{S}a$, following Chapter 8: Relations, Item 1 of Definition 8.1.3.1.1. Choosing $b=a$, we see that this condition is indeed satisfied. Thus $S\mathbin {\diamond }R$ is reflexive.
