Latter-day Saint Egyptologist Kerry Muhlestein discusses Joseph's involvement with the interpretation of the facsimiles in the Book of Abraham,

First, we must be clear that we do not know for sure that Joseph Smith authored the explanations of the facsimiles that were printed in the Times and Seasons, (on the acquisition of the papyri and publication of the Book of Abraham, see column 2,) which eventually became part of the Pearl of Great Price. While we do not know if Joseph Smith is the original author of these interpretations, we know he participated in preparing the published interpretations and gave editorial approval to them.

For example, on March 1, 1842, his journal records being at the printing office “correcting the first plate or cut of the records of father Abraham, prepared by Reuben Hedlock for the Times & Season” [spelling corrected]. The next day he wrote that he served for the first time as the editor of the Times and Seasons, reading through the proofs “in which is the commencement of the Book of Abraham.” On March 4th he worked again with Reuben Hedlock preparing the cut for the second facsimile. On March 9th he examined the copy of the Times and Seasons in which that facsimile would be published.[i] Thus, while the precise authorship of the explanations may be in question, the Prophet’s approval of them is not.[1]