A differentiable map $f:M\longrightarrow N$ induces pullback functions $f^*:\Omega^k(N)\longrightarrow \Omega^k(M)$. But since the identity $\mathrm{d}\circ f^*=f^*\circ\mathrm{d}$, we indeed have a complex homomorphism

that is,

which induces homomorphisms in cohomology

for each $k$. The usual properties $(f\circ g)^*=g^*\circ f^*$, $(\mathrm{id}_M)^*=\mathrm{id}_{H(M)}$ hold. Therefore a diffeomorphism $f:M\longrightarrow N$ induces isomorphisms in the cohomology groups