This imagery of three generations is a play on the different Greek verbs used in James 1:15
James 1 13 When tempted, no one should say, “God is tempting me.” For God cannot be tempted by evil, nor does he tempt anyone; 14 but each person is tempted when they are dragged away by their own evil desire and enticed. 15 Then, after desire has conceived, it gives birth to sin; and sin, when it is full-grown, gives birth to death. (NIV)
The Greek text here reads 1:15 εἶτα ἡ ἐπιθυμία συλλαβοῦσα τίκτει ἁμαρτίαν ἡ δὲ ἁμαρτία ἀποτελεσθεῖσα ἀποκύει θάνατον
The verb syllambanō can just mean “take”, but in the context of reproduction – for example with Elisabeth and Mary – it means conceive, become pregnant. Likewise tiktō “bring forth” means “give birth to” as from a woman. Straightforward.
However the second word rendered “give birth to” in some versions (like NIV above) in James 1:15 is better rendered “bring forth” or “produce” – as can be seen in James 1:18 where apokyeō is used a second time. And in the example below the RSV renders the word “bring forth” both for sin and for God, as a contrast of two different begettals.
15 Then desire when it has conceived gives birth to sin; and sin when it is full-grown brings forth death. 16 Do not be deceived, my beloved brethren. 17 Every good endowment and every perfect gift is from above, coming down from the Father of lights with whom there is no variation or shadow due to change.18 Of his own will he brought us forth by the word of truth that we should be a kind of first fruits of his creatures. (RSV)
These gives gender to desire (the grandmother), sin (the father, male), death (the grandchild)