Assumptions: binary products, equalizers
Conclusions: pullbacks
Reason: The pullback of f:X→Sf : X \to Sf:X→S and g:Y→Sg : Y \to Sg:Y→S is the equalizer of p1∘f, p2∘g:X×Y⇉Sp_1 \circ f, \, p_2 \circ g : X \times Y \rightrightarrows Sp1∘f,p2∘g:X×Y⇉S.