Left panels: regions of BH formation by direct collapse (in black), of NS formation (in orange) and of the coexistence of fallback BH and NS remnants (in grey) in the stellar parameter space spanned by M_CO and Z for each MT history class (single stars, Case C, Case B, and Case A donors), as predicted by the CCSN recipe introduced in this work. The M_CO range is limited to M_CO ∈ (M_CO^min, M_CO^max) (borderlines in red), outside of which only successful and failed SNe, respectively, are predicted to occur. For M_CO < M_CO⁽¹⁾, NS formation is guaranteed. For M_CO ∈ (M_CO⁽¹⁾, M_CO⁽²⁾), the CCSN outcomes are failed SNe, leaving behind BHs formed by direct collapse. For M_CO ∈ (M_CO⁽²⁾, M_CO⁽³⁾), successful SNe are guaranteed. Over this range, the compact remnant is a NS if M_CO ∈ (M_CO^NS, 1, M_CO^NS, 2). Otherwise, the remnant is a NS, at a probability of P = 0.85, or a fallback BH, at a probability of P = 0.15. The Z-dependence is modeled using M_CO⁽ⁱ⁾ ∝ log(Z/Z_⊙). Right panels: example of a statistical realization of the CCSN recipe with probabilistic fallback BH formation.