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Fix a partial function $a:Y\\to X$, and define the operation $\\star_a$ on $\\mathcal{PT}_{XY}$ by $f\\star_ag=fag$ for $f,g\\in\\mathcal{PT}_{XY}$. The sandwich semigroup $(\\mathcal{PT}_{XY},\\star_a)$ is denoted $\\mathcal{PT}_{XY}^a$. We apply general results from Part I to thoroughly describe the structural and combinatorial properties of $\\mathcal{PT}_{XY}^a$, as well as its regular and idempotent-generated subsemigroups, Reg$(\\mathcal{PT}_{XY}^a)$ and $\\mathbb E(\\mathcal{PT}_{XY}^a)$. 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