We will use without further mention that we've constructed our cycle class map $\gamma$ in Lemma \ref{lemma-cycle-classes} compatible with intersection products and pullbacks and that we've already shown axioms (A), (B), (C)(a), (C)(c), and (C)(d) of Section \ref{section-axioms}, see Lemma \ref{lemma-poincare-duality}, Remark \ref{remark-trace}, and Lemmas \ref{lemma-trace-product} and \ref{lemma-trace-base}. In particular, we may use (for example) Lemma \ref{lemma-pushforward} to see that pushforward on $H^*$ is compatible with composition and satisfies the projection formula. \medskip\noindent Let $f : X \to Y$ be a morphism of nonempty equidimensional smooth projective schemes over $k$. We are trying to show $f_*\gamma(\alpha) = \gamma(f_*\alpha)$ for any cycle class $\alpha$ on $X$. We can write $\alpha$ as a $\mathbf{Q}$-linear combination of products of Chern classes of locally free $\mathcal{O}_X$-modules (Chow Homology, Lemma \ref{chow-lemma-K-tensor-Q}). Thus we may assume $\alpha$ is a product of Chern classes of finite locally free $\mathcal{O}_X$-modules $\mathcal{E}_1, \ldots, \mathcal{E}_r$. Pick $p : P \to X$ as in the splitting principle (Chow Homology, Lemma \ref{chow-lemma-splitting-principle}). By Chow Homology, Remark \ref{chow-remark-the-proof-shows-more} we see that $p$ is a composition of projective space bundles and that $\alpha = p_*(\xi_1 \cap \ldots \cap \xi_d \cap \cdot p^*\alpha)$ where $\xi_i$ are first Chern classes of invertible modules. By Lemma \ref{lemma-ok-for-projective-bundle} we know that $p_*$ commutes with cycle classes. Thus it suffices to prove the property for the composition $f \circ p$. Since $p^*\mathcal{E}_1, \ldots, p^*\mathcal{E}_r$ have filtrations whose successive quotients are invertible modules, this reduces us to the case where $\alpha$ is of the form $\xi_1 \cap \ldots \cap \xi_t \cap [X]$ for some first Chern classes $\xi_i$ of invertible modules $\mathcal{L}_i$. \medskip\noindent Assume $\alpha = c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_t) \cap [X]$ for some invertible modules $\mathcal{L}_i$ on $X$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_X$-module. For $n \gg 0$ the invertible $\mathcal{O}_X$-modules $\mathcal{L}^{\otimes n}$ and $\mathcal{L}_1 \otimes \mathcal{L}^{\otimes n}$ are both very ample on $X$ over $k$, see Morphisms, Lemma \ref{morphisms-lemma-invertible-add-enough-ample-very-ample}. Since $c_1(\mathcal{L}_1) = c_1(\mathcal{L}_1 \otimes \mathcal{L}^{\otimes n}) - c_1(\mathcal{L}^{\otimes n})$ this reduces us to the case where $\mathcal{L}_1$ is very ample. Repeating this with $\mathcal{L}_i$ for $i = 2, \ldots, t$ we reduce to the case where $\mathcal{L}_i$ is very ample on $X$ over $k$ for all $i = 1, \ldots, t$. \medskip\noindent Assume $k$ is infinite and $\alpha =  c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_t) \cap [X]$ for some very ample invertible modules $\mathcal{L}_i$ on $X$ over $k$. By Bertini in the form of Varieties, Lemma \ref{varieties-lemma-bertini} we can successively find regular sections $s_i$ of $\mathcal{L}_i$ such that the schemes $Z(s_1) \cap \ldots \cap Z(s_i)$ are smooth over $k$ and of codimension $i$ in $X$. By the construction of capping with the first class of an invertible module (going back to Chow Homology, Definition \ref{chow-definition-divisor-invertible-sheaf}), this reduces us to the case where $\alpha = [Z]$ for some nonempty smooth closed subscheme $Z \subset X$ which is equidimensional. \medskip\noindent Assume $\alpha = [Z]$ where $Z \subset X$ is a smooth closed subscheme. Choose a closed embedding $X \to \mathbf{P}^n$. We can factor $f$ as $$ X \to Y \times \mathbf{P}^n \to Y $$ Since we know the result for the second morphism by Lemma \ref{lemma-ok-for-projective-bundle} it suffices to prove the result when $\alpha = [Z]$ where $i : Z \to X$ is a closed immersion  and $f$ is a closed immersion. Then $j = f \circ i$ is a closed embedding too. Using the hypothesis for $i$ and $j$ we win. \medskip\noindent We still have to prove the lemma in case $k$ is finite. We urge the reader to skip the rest of the proof. Everything we said above continues to work, except that we do not know we can choose the sections $s_i$ cutting out our $Z$ over $k$ as $k$ is finite. However, we do know that we can find $s_i$ over the algebraic closure $\overline{k}$ of $k$ (by the same lemma). This means that we can find a finite extension $k'/k$ such that our sections $s_i$ are defined over $k'$. Denote $\pi : X_{k'} \to X$ the projection. The arguments above shows that we get the desired conclusion (from the assumption in the lemma) for the cycle $\pi^*\alpha$ and the morphism $f \circ \pi : X_{k'} \to Y$. We have $\pi_*\pi^*\alpha = [k' : k] \alpha$, see Chow Homology, Lemma \ref{chow-lemma-finite-flat}. On the other hand, we have $$ \pi_*\gamma(\pi^*\alpha) = \pi_*\pi^*\gamma(\alpha) = \gamma(\alpha) \pi_*1 $$ by the projection formula for our cohomology theory. Observe that $\pi$ is a projection (!) and hence we have $\pi_*(1) = \int_{\Spec(k')}(1) 1$ by Lemma \ref{lemma-pr2star}. Thus to finish the proof in the finite field case, it suffices to prove that $\int_{\Spec(k')}(1) = [k' : k]$ which we do in Lemma \ref{lemma-integrate-1}.