For the phoneme /dʒ/, one can argue that the Cree and French components behave differently. For the French elements, it is part of the normal phoneme inventory, and appears initially and medially (dzhis 'ten', midzhii 'noon') but not word-finally. Historically, these affricates are all the result of palatalization of /d/ before high front vowels.
For Cree elements, it is only found in specific environments, and only word-initially:
(1) initially in verbs. Here a verb starting in /ch/ has merged with the prefix ni- into /dʒ/, as in dzhikahikaan 'I am chopping things', from the verb chikahikee- 'to chop things'.
(2) initially in possessive nouns dzhahkush 'my sister-in-law'.