In Goodwillie's "Calculus I", speaking of a commutative diagram of spaces

$$\begin{array}{c} Y & \rightarrow & Y_1 \\ \downarrow & & \downarrow & \\ Y_2 & \rightarrow & Y_{12} \end{array}$$

there is the following statement

'Cartesian' implies that (for every basepoint in $Y$) the relative homotopy groups of $Y \rightarrow Y_1$ map isomorphically to those of $Y_2 \rightarrow Y_{12}$. Similarly 'co-Cartesian' implies that the relative homology groups of $Y \rightarrow Y_1$, map isomorphically to those of $Y_2 \rightarrow Y_{12}.$

Here Cartesian (resp. co-Cartesian) means that $Y$ ($Y_{12}$) is equivalent via the canonical map to the homotopy limit (homotopy colimit).

What is a more precise sense in which the two parts of the statement about homotopy and homology are similar to each other?