Singular chains are covariantly functorial

Define $f_{\mathrm{\#}}(\sum a_{\sigma }[\sigma ])=\sum a_{\sigma }[f\circ \sigma ]$. Composites are continuous, the sum is finite, and collecting equal images preserves addition and real scalar multiplication. Under the real tensor identification this is exactly the map in [F2]. In negative degrees it is the unique map between zero spaces.

For $k\ge 1$, $\mathrm{\partial }{f}_{\mathrm{\#}}[\sigma ]={\sum }_{i=0}^k(-1{)}^i[f\circ \sigma \circ {\delta }_i]=f_{\mathrm{\#}}\mathrm{\partial }[\sigma ]$. Real linearity extends equality to all chains; for $k\le 0$ both composites are zero. This includes constant and degenerate simplices because the computation does not discard any face.

On every generator, $g_{\mathrm{\#}}{f}_{\mathrm{\#}}[\sigma ]=[g\circ f\circ \sigma ]=(g\circ f{)}_{\mathrm{\#}}[\sigma ]$ and $({\mathrm{id}}_X{)}_{\mathrm{\#}}[\sigma ]=[\sigma ]$. Finite linear extension proves both laws. They also hold on zero groups, including all chains of the empty space. On a point each nonnegative chain map induced by its identity is the identity of $\mathbb{R}$. All maps are given by formulas on supplied generators; no AC is used.