  We investigate pairing fluctuation effects in a two band fermionic system,
where a shallow band in the Bardeen--Cooper--Schrieffer--Bose--Einstein
condensation (BCS-BEC) crossover regime is coupled with a weakly interacting
deep band. Within a diagrammatic $T$ matrix approach, we~report how
thermodynamic quantities such as the critical temperature, chemical potential,
{and~momentum distributions} undergo the crossover from the BCS to BEC regime
by tuning the intraband coupling in the shallow band. We also generalize the
definition of Tan's contact to a two band system and report the two contacts
for different pair-exchange couplings. The present results are compared with
those obtained by the simpler Nozi\`eres--Schmitt--Rink approximation. We
confirm a pronounced enhancement of the critical temperature due to the
multiband configuration, as well as to the pair-exchange coupling.
