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For a general $\lambda$-deformation of current algebra CFTs we compute theexact Weyl anomaly coefficient and the corresponding metric in the couplingsspace geometry. By incorporating the exact $\beta$-function found in previousworks we show that the Weyl anomaly is in fact the exact Zamolodchikov's$C$-function interpolating between exact CFTs occurring in the UV and in theIR. We provide explicit examples with the anisotropic $SU(2)$ case presented indetail. The anomalous dimension of the operator driving the deformation is alsocomputed in general. Agreement is found with special cases existing already inthe literature.

Weyl anomaly and the $C$-function in $\lambda$-deformed CFTs