Fillers in span factorization system

Remark 17.1.11. The unique filler in a lifting problem in \(\Span _{L,R}(C)\) can be visualized directly. A commutative square with a backwards map on the left and a forwards map on the right corresponds to diagram of spans in \(C\) as follows:

Here the composite span is \(A \leftarrow Z' \rightarrow Y\), computed both as the composite of the spans \(A \xleftarrow {l} B \xrightarrow {\id _B} B\) and \(B \leftarrow Z''\to Y\), as well as the composite of the spans \(A \leftarrow Z \to X\) and \(X \xleftarrow {\id _X} X \xrightarrow {r} Y\). The unique filler span is then given by the span \(B \leftarrow Z' \rightarrow X\).