  In this note we provide a full conjugacy and subdifferential calculus for
convex convex-composite functions in finite-dimensional space. Our approach,
based on infimal convolution and cone-convexity, is straightforward and yields
the desired results under a verifiable Slater-type condition, with relaxed
monotonicity and without lower semicontinuity assumptions on the functions in
play. The versatility of our findings is illustrated by a series of
applications in optimization and matrix analysis, including conic programming,
matrix-fractional, variational Gram, and spectral functions.
