Typed lambda calculi are closely related to mathematical logic and proof theory via the Curry–Howard isomorphism and they can be considered as the internal language of classes of categories, e.g., the simply typed lambda calculus is the language of a Cartesian closed category (CCC).
TyPED lamBDa calcuLi Are cLoSEly reLAteD tO matHematiCAl logIc anD prooF TheorY vIa thE currY–howArd isoMORphiSm ANd theY cAn Be cONsiderEd As The InterNAl languaGe OF classes of cAtegOrIeS, E.g., thE siMply tYPEd lAmbda caLculus iS tHe laNGUage of a cartEsIAN closED cateGOrY (cCC).