update with the group representive value

[joon]

original data

 
(ID) [category] {name}
(1)    [1]          {Tom}
(2)    [1]          {Jane}
(3)    [3]          {Kate}
(4)    [1]          {Ted}
(5)    [3]         {Mary}
(6)    [6]         {John}
(7)    [6]         {Juno}
(8)    [8]         {Emily}
(9)    [6]         {Ross}

I have the above data in myTable.

I can group the above data by min(category) as min_category .

It will produce

group by category

(1) [1] {Tom}
(3) [3] {Kate}   

(6) [6] {John}
(8) [8} {Emily}

Now I'll tell you what I want.
I like to update all the name values in a same group with the group representitive value.

target data

(ID) [category] {name}
(1)    [1]          {Tom}
(2)    [1]          {TOM}
(3)    [3]          {Kate}
(4)    [1]          {TOM}
(5)    [3]         {KATE}
(6)    [6]         {John}
(7)    [6]         {JOHN}
(8)    [8]         {Emily}
(9)    [6]         {JOHN}

How can I make the target data from the original data?

Thanks in Advance