Let $ \mathcal{E} (k) $ be the enumerative category of smooth projective varieties over a field $k$. This is a $ \mathbb{Q} $ - linear category which has for objects : smooth projective varieties over $k$, and for morphismes the algebraic correspondances of degree $ 0 $ modulo the numerical equivalence as follows :
$$ \mathcal{E} (k) (X,Y) = \{ \ \text{ algebraic cycles ( with rational coefficients ) of codimension } \ \mathrm{dim} X \\ \mathrm{over} \ X \times Y \ \} / \sim_{ \mathrm{num} } $$
In a textbook which i'm learning right now, the author states that :
The composition of morphisms of $ \mathcal{E} (k) (X,Y) $ is given by the following law : $$ g \circ f = ( \mathrm{pr}_{XZ}^{XYZ} )_* \big( ( \mathrm{pr}_{XZ}^{XYZ} )^* . \mathrm{pr}_{YZ}^{XYZ} )^* g \big) $$ Could you tell me why is : $$ g \circ f = ( \mathrm{pr}_{XZ}^{XYZ} )_* \big( ( \mathrm{pr}_{XZ}^{XYZ} )^* . \mathrm{pr}_{YZ}^{XYZ} )^* g \big) $$ ?
Thank you.