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 $Y\_2 \rightarrow Y\_{12}.$
> 
> 
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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? 



Homotopy pullbacks/relative homotopy groups vs homotopy pushouts/relative homology groups
Relative homotopy groups are the homotopy groups of the homotopy fiber. A homotopy pullback square induces an equivalence of the homotopy fibers of two of its parellel maps by the cancellation property of homotopy pullbacks.


Similarly, relative homology groups are the homology groups of the homotopy cofiber. The statement follows by a dual argument.


