The walking reflexive pair has two objects $0,1$, the identities, the morphisms $f,g : 0 \rightrightarrows 1$, $r : 1 \to 0$ with $fr = gr = \mathrm{id}_1$, and no other relations. Then we have two (split) idempotent morphisms $rf, rg : 0 \rightrightarrows 0$, and they satisfy $rf \circ rg = rg$, $rg \circ rf = rf$, $f \circ rf = f$, $g \circ rf = f$, $f \circ rg = g$, $g \circ rg = g$, $rf \circ r = r$, $rg \circ r = r$. Based on this I think that there are 7 morphisms in total.
Let's add this category to the database and decide all of its properties. Related categories in the database are the walking parallel pair and the walking idempotent.
The category has already been used in the proof of this result.
Related: #123
The walking reflexive pair has two objects$0,1$ , the identities, the morphisms $f,g : 0 \rightrightarrows 1$ , $r : 1 \to 0$ with $fr = gr = \mathrm{id}_1$ , and no other relations. Then we have two (split) idempotent morphisms $rf, rg : 0 \rightrightarrows 0$ , and they satisfy $rf \circ rg = rg$ , $rg \circ rf = rf$ , $f \circ rf = f$ , $g \circ rf = f$ , $f \circ rg = g$ , $g \circ rg = g$ , $rf \circ r = r$ , $rg \circ r = r$ . Based on this I think that there are 7 morphisms in total.
Let's add this category to the database and decide all of its properties. Related categories in the database are the walking parallel pair and the walking idempotent.
The category has already been used in the proof of this result.
Related: #123