circuits – For a logic gate to be universal, must it necessarily be able to perform duplication?

It is said that a gate that can simulate AND and NOT is universal and able to recreate any classical circuit. I was looking at some of the circuits simulated by NAND, and for some of them, we need to split the output of a gate and direct each of the splits to other NAND gates (see the area circled in red):
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If we make the restriction that all gates are reversible (in particular, we require fixed input/output size), does the AND/NOT criterion still hold? Is the ability to “split” outputs (i.e., a gate that can do AND, NOT, and duplicate a $0$ or $1$, e.g. $(0,0,1) rightarrow (0,1,1)$ ) a necessary/sufficient condition for universality?